BIOLOGICAL ION CHANNELSBIOLOGICAL ION CHANNELSAS NANOSCALE DEVICESAS NANOSCALE DEVICES

Approaches to Simulation:Continuum and Particle Methods

T.A. van der Straaten

Beckman Institute for Advanced Science and TechnologyUniversity of Illinois at Urbana-Champaign

Department of Molecular Biophysics and PhysiologyRush Medical College, Chicago

OUTLINEOUTLINE

• Background on Protein chemistry, Ion Channels etc

• Simulation Methods

• Continuum (Drift-Diffusion) simulations

– gramicidin

– porin

• Monte-Carlo simulations – gramicidin

Proteins that form nanoscopic aqueous tunnels in cell membranePhysiological Role – regulate ion flow and composition inside cell control,electrical signaling in the nervous system, muscle contraction, drug deliveryDisease – malfunctioning channels

BIOLOGICAL ION CHANNELSBIOLOGICAL ION CHANNELS

Amino acids – building blocksof proteins

Side-chain distinguishes theamino acid

Some of the side-chains areionizable – proteins are highlycharged. Strong and steeplyvarying charge density iscritical to the I-V characteristicsof the open channel.

neutral pH: NH2 → NH3+

COOH → COO-

Amino acids are linked togetherby peptide bonds

Polypeptide chains fold to formproteins

BIOCHEMISTRY OF ION CHANNELSBIOCHEMISTRY OF ION CHANNELS

NH2 — Cα — Cβ — OH

RR

H

|

| ||Oamino

carboxylside-chain

N — Cα — Cβ —

H

RR22

|

|

||ORR11

H

|

| ||O

— N — Cα — Cβ —

H|

H|

H2Opeptide bond

PROTEIN FOLDINGPROTEIN FOLDINGIonic bondsHydrogren bondsvan der Waals attraction

3D structure is determined byorder of amino acids in sequenceand energy considerations -folded structure is that whichminimizes free energy

Conformational changes canoccur when protein interacts withother molecules - crucial tofunction of protein

electrostatic

GRAMICIDIN

Small simple channel forming moleculeEach monomer is made up of 15 amino acids (~ 500 atoms) folded intoa helical structure.Expressed by certain bacteria perhaps to kill other microorganisms bycollapsing the ion gradients that are required for their survival. Useful asantibiotic

ompF PORIN

Large trimeric channel, sits in the outer membrane of e. coliEach monomer is made up of 340 amino acids ( ~90 000 atoms)

RASMOL www.umass.edu/microbio/rasmol/PROTEIN DATABANK www.rcsb.org/pdb/

GATING / SWITCHINGGATING / SWITCHING

Many channels exhibit switching properties similar to electronic devices.

“open/close” or “on/off” states – response to environment

100 mM KCl : x mM KCl

x = 1000

x = 500

-400

-200

0

200

400

Trim

er C

urre

nt (p

A)

-200 -100 0 100 200

Vbias(mV)

channel closedompF porin

PATCH-CLAMP MEASUREMENTSPATCH-CLAMP MEASUREMENTS

Allows single channel currents to be recorded

Channel is either fully open or fully closed

Open channel conductance is constant - aggregate current reflectsthe total number of channels open at any given time

Allows single channel currentsto be recorded

Channel is either fully open orfully closed

Open channel conductance isconstant - aggregate currentreflects the total number ofchannels open at any giventime

PATCH-CLAMP MEASUREMENTSPATCH-CLAMP MEASUREMENTS

GATING MECHANISMSGATING MECHANISMS

Voltage Gating – the open probability is a function of voltage – e.g., Na+,K+, and Ca2+ channels are essential for conducting a nerve pulse down anaxon and to another nerve cell (or neuron).

pH Gating –open probability is a function of pHLigand Gating – small molecule binding to the channel affects its openprobability. Important in chemical synaptic transmission (the mostcommon way of transfering a signal from one neuron to another). Thesechannels are gated by neurotransmitters, molecules that actually carrythe signal between two nerve cells.

Mechanical Gating – channel is directly gated by a mechanical trigger–e.g., cation channel in the hair cell of the inner ear, which is directlygated by a mechanical vibration caused by sound. Not well studiedbecause of technical difficulties

GRAMICIDIN FORMS DIMERSGRAMICIDIN FORMS DIMERS

open closed

lipidbilayer

curr

ent (

pA)

time (s)

singlechannel

twochannels

threechannels

KcsA KcsA CHANNEL - a natural pH sensorCHANNEL - a natural pH sensorpH induced conformational changes causes KcsA channel to switchbetween conducting and non-conducting states

CLOSED OPEN

CLOSED OPEN

(Jay Mashl, computational biology)

Changes in pH → protein charge → electrostatic fields → conformationalchanges → lowering energy barrier to permeation

SELECTIVITYSELECTIVITYMany channels selectively transmit or block a particular ion species

• Gramicidin passes only small monovalent positive ions (e.g., H+, Li+,Na+, K+) – electrostatic and steric barriers• Porin ompF channel shows a mild preference for cations• Potassium channel selects K+ over Na+ by a factor of 104, even thoughthese ions are similar in size – dehydration of Na+ presents an energybarrier

Currently only ~ 50 known channel structures~ 30% of all the genes in the human genome code for ion channels

Device feature sizes are shrinking, how much further can we go?

Channels are naturally occurring device elements– self-assembled– perfectly reproducible– have many specific built-in features and functions– can be mutated

Design channels with specific conductances, selectivities and functions.Bio-devices – circuit elements, biosensors

ENGINEERING APPLICATIONSENGINEERING APPLICATIONS

AMBRIAMBRI®® BIOSENSOR BIOSENSOR

- Chemical to bedetected binds to theantibodies

- Prevents theformation of current-conducting dimers

- Marked reductionin aggregate current

www.ambri.com.au

Crystal bandstructureCarriers are quasi-particleswith a small effective mass

Conduction channel is delimited bydepletion layers or potential energy

steps at heterojunctionsUltra-scaled structures suffer from

fluctuations of doping and sizes

Energetic carriers may causestructural damage

Water solution.Carriers are ions with relativelylarge mass and finite volumeConduction channel is delimited bya highly charged protein membrane

Channel structures are alwaysperfect replicas. Stable mutationscan be used to modify channel behavior.Ion are thermalized by interactionwith water.

ION CHANNELS AS NANODEVICESION CHANNELS AS NANODEVICES

solid state ion channels

Permeation takes place on timescales of the order of millisecondsScattering rates are high (1013 - 1014 Hz)

System length scales ~ 100 ÅFeature length scales ~ 1 Å (steep gradients)Complicated, temporally fluctuating structures, difficult to mesh

Representation of trace ionic concentrations

Dielectric constant - protein? lipid? inside channel?Ion-water scattering rates? Other types of interaction with water?Water behavior at the nanoscale?

COMPUTATIONAL ISSUESCOMPUTATIONAL ISSUES

IRREGULAR PROTEIN GEOMETRYIRREGULAR PROTEIN GEOMETRY

topology and charge varies on scale-length of 1Åmolecular “surface” rendered with GRASP

~50Å

SIMULATION HIERARCHYSIMULATION HIERARCHY

Molecular Dynamics

Brownian Dynamics

Monte-Carlo Models

Continuum Models

Compact ModelsCircuit Models

Partial charges

Transport,correctionsfor ion size

IV curve

Quantum Chemistry

Transportparameters

O(N2) N~105 ∆t~1fs Tsim~1-10nsCPU weeks-months

Single amino acid (10 to 20 atoms), no dynamics

O(M) M~104 N~100 ∆t~10fs Tsim~100ns CPU days-weeks

Langevin Eq, overdamped limit, ∆t~100fsTsim~0.1-1µs CPU days, problems with ε(r)

full IV curve CPU mins-hours, overestimatesion densities, requires transport parameters

BECKMAN INSTITUTEBECKMAN INSTITUTE

• combine computational electronics and computationalbiochemistry tools to simulate the behavior of ionicchannels as devices.

• study how ion channels could be incorporated intraditional electronic systems performing functions, andlearn how to simulate wet/dry interfaces.

• understand how specialized functions are accomplishedby natural channels, and investigate how to embed similarfunction in synthetic channels.

• exploit self-assembly feature of molecular structure toarrive at new device architectures.

( / )j D kT ρ φ± ± ± ±

= − ∇

DRIFT-DIFFUSIONDRIFT-DIFFUSION

ln exq kTφ ψ ρ φ±±±

= ± +

Poisson’s Equation

Drift-diffusion Equation

Electrochemical potential

∇•(ε∇ψ) = − ( ρfixed + ρ+ + ρ–)

j St

ρ±

± ±

∂∇ + =

∂i Continuity Equation

right left biasVψ ψ− =

, right rightCρ

±=

, left leftCρ

±=

Boundary Conditions

• PROPHET – a rapid prototyping computational environment developedat Lucent Technologies (Conor Rafferty) for solving multi-dimensionalsystems of PDEs.

• Dial-an-Operator scripting language allows user to construct a systemof equations at an abstract level, using standard ‘building blocks’ (e.g.,Laplacian operator).

• Database hierarchy for numerical and physical parameters with‘inheritance’ features – data can be created or modified ‘on the fly’.

• Physical models can be developed independently of the details of thediscretization library, solvers, grid structure, etc.

• Handles arbitrary multi-dimensional geometries and physical systems.

PROPHET SIMULATORPROPHET SIMULATOR

Application of PROPHET to Ion ChannelsApplication of PROPHET to Ion Channels

• The Stanford group (R.W. Dutton, Z.Yu, D.Yergeau) aredeveloping the new generation of PROPHET with extensions togeneral device simulations.

• A three-way collaboration (UIUC, Stanford, Rush) is underway toapply PROPHET to simulate ion permeation through protein ‘nano-channels’ embedded in cell membranes, using a continuum transportmodel

• Detailed protein geometry and fixed charge distribution areconstructed using a variety of existing molecular biology andchemistry software tools (UHBD, GROMOS). This structuralinformation is then translated into PROPHET native mesh formatusing a customized mesh generator.

Generate customizedPROPHET mesh

Define protein surfaceflag grid points (UHBD)in protein, lipid bilayer

Download proteinstructure

(protein databank)atomic coordinates, radiiamino acid sequence

Assign chargeto each atomic

coordinate

Interpolatecharge to grid

ρfixed

PROPHET script–Specify solver parameters–Input physical domain–Assemble PDE system–Input BCs & physical parameters–Solve (Newton’s method)–Write output

Postprocessorreconstruct j± and I

current to electrodes.

D, µfluctuation

analysis of MDion trajectories

(GROMOS)

PROTEIN STRUCTURE PROTEIN STRUCTURE →→ IV CURVEIV CURVE

dbase prefix=library/math/systems/default_numerical_parametersdbase create name=method sval=iterativedbase create name=maxNewton ival=50dbase create name=accel sval=bicgdbase create name=precon sval=iludbase create name=maxfil ival=1dbase create name=dynfil ival=0dbase create name=NewtonMaxUpd rval=1.0e11

dbase prefix=""dbase create name=/options/ignoreFPE ival=0dbase create name=library/math/memory.size ival=770

dbase createlist name=library/physics/channeldbase create name=library/physics/channel/max.process.temperature rval=1500dbase createlist name=library/physics/proteindbase create name=library/physics/protein/max.process.temperature rval=1500dbase createlist name=library/physics/lipiddbase create name=library/physics/lipid/max.process.temperature rval=1500

load pas=/home/amiga/trudy/prophet_examples/porin_NLP.pas

boundary name=left+ xmin=-48.0 xmax=48.0+ ymin=-48.0 ymax=48.0+ zmin=-48.0 zmax=-48.0

boundary name=right+ xmin=-48.0 xmax=48.0+ ymin=-48.0 ymax=48.0+ zmin=48.0 zmax=48.0

specify solverparameters

input domain

define electrodes

system name=pnp+ sysvars=K,Cl,phi+ nterm=10## Solve transport in channel only#+ term0=box_div.sg_ddTherm(phi,K,tl,K_mobility|K)@{channel}+ term1=box_div.sg_ddTherm(phi,Cl,tl,Cl_mobility|Cl)@{channel}## Dirichlet B.C. for carrier densities at left & right boundaries#+ term2=dirichlet.default_dirichlet(0|K)@{channel/left,channel/right}+ term3=dirichlet.default_dirichlet(0|Cl)@{channel/left,channel/right}## Solve Poisson everywhere#+ term4=-1*box_div.lapflux(phi|phi)@{channel,protein,lipid}+ term5=nodal.copy(K|phi)@{channel}+ term6=-1*nodal.copy(Cl|phi)@{channel}+ term7=nodal.copy(rho_fixed|phi)@{protein,lipid}## Dirichlet B.C. for phi at left & right boundaries#+ term8=dirichlet.default_dirichlet(0|phi)@{channel/left,channel/right}## constrain phi to be continuous across channel/protein/lipid interfaces#+ term9=constraint.continuity(phi|phi)@{channel/protein,protein/lipid,channel/lipid}

assemble PDE system

# define the K in channel #

dbase createlist name=library/physics/channel/Kdbase prefix=library/physics/channel/Kdbase create name=background rval=1.0e-5dbase create name=Cstar rval=1 # ?dbase create name=esign rval=1 # ?dbase create name=scale rval=1.0e-5 # ?dbase create name=dirichlet.left rval=0.602214e6 # 100 mMdbase create name=dirichlet.right rval=0.602214e6 # 100 mMdbase prefix=""

... similarly for Cl ...

# define the phi in channel #

dbase createlist name=library/physics/channel/phidbase prefix=library/physics/channel/phidbase create name=scale rval=1.0e-7dbase create name=background rval=0.0dbase create name=Cstar rval=1dbase create name=Dix rval= 0.4425e10 # Dix = eps/q eps = 80*eps0dbase create name=dirichlet.left rval=0.0 # zero biasdbase create name=dirichlet.right rval=0.0dbase prefix=""

... similarly for phi in protein and lipid ...

physical parameters &boundary conditions

field set=rho_fixed profile3d="porin_charge"field set=K_mobility profile3d="K_mobility"field set=Cl_mobility profile3d="Cl_mobility"field set=tl val=300

# dump out flux datafield set=flux solvar=K type=edgefield set=flux solvar=Cl type=edge

# print out the Newton convergence behaviordbase create name=/options/loops ival=1

solve system=pnpsave pas=porin.0mV.pas

dbase prefix=library/physics/channel/phidbase modify name=dirichlet.left rval=0.0dbase modify name=dirichlet.right rval=0.01 # bias 10 mVdbase prefix=""solve system=pnpsave pas=porin.10mV.pas

physicalparameters

solve system &write output

Lipid εr = 2

Protein εr = 20

Solution εr = 80bath (NaCl)

x

z

y

Slice through the channel ofthe 3D mesh used inPROPHET. (48x48x64 gridpoints, 0.5Å mesh spacing)

GRAMICIDIN

Å

Potential distribution, on aslice throught the channel.

Large positive and negativepotential spikes in the proteinregion are due to the fixedcharges.

GRAMICIDIN1.0 molar NaCl solution

+Vbias = 150 mV

Volts

Å

GRAMICIDIN3-D PROPHET SIMULATION

Cl− Anion distribution

BIAS = 150 mV

Bath solution = 1.0 molar NaCl

Å

6.02 × 106 = 1 molar

GRAMICIDIN3-D PROPHET SIMULATION

Na+ Cation distribution

BIAS = 150 mV

Bath solution = 1.0 molar NaCl

Å

6.02 × 106 = 1 molar

ompFompF ( (OOUTER UTER MMEMBRANE EMBRANE PPORIN)ORIN)

• Trimeric protein that resides in theouter membrane of e-coli bacterium.

• Well-known, very stable structure

• Net charge of ~ -30|e| Highly charged pore constriction. Moderate cation selectivity.

• Unknown gating mechanism

• Can be mutated. Ideal for experimental and simulation studies of ion permeation. Template for biodevices

electrodes

lipid (ε=2)

protein (ε=20)

96Å

96Å

electrolyte (ε=80)

96Å

lipid (ε=2)

protein (ε=20)

aqueouspores

(ε=80)

side view cross-section

PROPHET MESH - PORINPROPHET MESH - PORIN

Molecular Dynamics↓

Fluctuation Analysis

Electrostatics +Statistical physics

↓Ionization states

Drift-diffusion↓

Ion Transport

Transportparameters

ComputationalComputationalBiochemistryBiochemistry

Proteincharges

ComputationalComputationalElectronicsElectronics

Current-Voltage Curves

Atomistic

Continuum

COMPUTATIONAL BIOCHEMISTRYCOMPUTATIONAL BIOCHEMISTRY

CALCULATING PROTEIN CHARGESCALCULATING PROTEIN CHARGES((Jakobsson Jakobsson group)group)

Amino acids …Side chains are ionizable Pion

Pion depends on local electric field, salt concentration and pH

How to compute Pion inside the folded protein?1. ∆(Free Energy): amino acid in H2O→ amino acid in protein Nonlinear Poisson (Poisson-Boltzmann) equation (UHBD)2. Effect of interaction of amino acids with each other

NH2 — C — COOH

RR

H

|

|

IONIZATION STATES FOR IONIZATION STATES FOR ompFompF PORIN PORIN

MOLECULAR DYNAMICS MOLECULAR DYNAMICS (Jakobsson group)

• System is represented bycharged balls (atoms)connected by springs (bonds)

• System is initialized with aMaxwellian distribution andallowed to evolve accordingto Newtonian mechanics.

• Gromacs - efficient, scalableopen source MD package

hydrocarbonmembrane

electrolyte

porin

τ

o }pore constriction

z = 23.1

z = 27.1

z = 19.1

z = 33.1

z = 37.1

Ao

Ao

Ao

Ao

Ao

0.0 5.0 10.0 15.0 20.0

[ps]

0.00

0.40

0.80

1.20

1.60

MSD

[A2 ]

( ) ( )( )2MSD( ) 6x t x t Dτ τ τ= + − =

ESTIMATING DIFFUSIVITY FROMESTIMATING DIFFUSIVITY FROMION TRAJECTORIESION TRAJECTORIES

DIFFUSIVITY INSIDE CHANNELDIFFUSIVITY INSIDE CHANNEL

DK+(z)

DCl-(z)

bulk KCl

0 20 40 60 80

Position along channel (Angstroms)

0.00

0.50

1.00

1.50

2.00

2.50

D (1

0-5 c

m2 /

sec)

channel

polynomial fit

3-level average

CURRENT-VOLTAGE CURVESCURRENT-VOLTAGE CURVESexperiment (Rush) vs. simulation (UIUC)

experiment

-150

-100

-50

0

50

100

150

200

Trim

er c

urre

nt (p

A)

-200 -150 -100 -50 0 100 150 200

Vbias(mV)

50

ompF 100mM KCl

polynomial fit

3-level average

D=0.67x10-5cm2/sec

Trim

er C

urre

nt (p

A)

Vbias(mV)

Cright=0.5M D=0.65x10-5cm2/sec

Cright=0.25M D=0.64x10-5cm2/sec

Cright=1M D=0.63x10-5cm2/sec

Cright=3M D=0.61x10-5cm2/sec

ompF (Cleft=0.1M KCl )

-400

-200

0

200

400

600

-200 -150 -100 -50 0 50 100 150 200

ASYMMETRIC BATH CONCENTRATIONSASYMMETRIC BATH CONCENTRATIONS

Vbias(mV)

Trim

er C

urre

nt (p

A)

-200 -150 -100 -50 0 50 100 150 200-1000

-500

0

500

1000ompF (Cleft=0.25M KCl )

Cright=0.1M D=0.74x10-5cm2/sec

Cright=0.5M D=0.67x10-5cm2/sec

Cright=1M D=0.67x10-5cm2/sec

Cright=3M D=0.67x10-5cm2/sec

ASYMMETRIC BATH CONCENTRATIONSASYMMETRIC BATH CONCENTRATIONS

Trim

er C

urre

nt (p

A)

Vbias(mV)

ompF (Cleft=0.5M KCl )

-1000

-500

0

500

1000

-200 -150 -100 -50 0 50 100 150 200

Cright=0.25M D=0.58x10-5cm2/sec

Cright=0.1M D=0.66x10-5cm2/sec

Cright=1M D=0.60x10-5cm2/sec

Cright=3M D=0.60x10-5cm2/sec

ASYMMETRIC BATH CONCENTRATIONSASYMMETRIC BATH CONCENTRATIONS

STUDYING MUTATIONS WITH PROPHETSTUDYING MUTATIONS WITH PROPHET

OmpF G119D

OmpF → G119D: replace an uncharged glycine G119 (white), located in the poreconstriction, with an aspartate D119 (red).

(visualization – VMD: http://www.ks.uiuc.edu/Research/vmd/)

D113

E117

G119

R42R82

R132

D113

E117D119

R42R82

R132

EQUILIBRIUM KEQUILIBRIUM K++ DENSITY DENSITY

ompF Mutation G119D

>0.75M

0M

100 mM: D = 6.78 x10-6 cm2s-1

1 M: D = 5.98x10-6 cm2s-1

1 M: (uncharged ASP119)

-400

-200

0

200

400

600

Trim

er c

urre

nt (p

A)

-200 -150 -100 -50 50 100 150 2000

Vbias (mV)

Effect of mutation: narrower channel, lower conductance (15-40% lower),higher cation selectivity

Separate geometric & electrostatic effects of the mutation by scaling downthe charge distribution on the aspartate to zero - ‘virtual’ channel that isstructurally identical to G119D but has the same charge as ompF.

Simulations at1M reveal nochange in theconductance.

Cation selectivityreduced

CONTINUUM SIMULATIONS – SUMMARYCONTINUUM SIMULATIONS – SUMMARY

• Combine existing methods of computational biochemistryand computational electronics to study ion channels.

• Current-voltage characteristics computed for ompF in KClagree reasonably well with experiments.

• Conductance sensitive to the diffusion coefficient profile,especially in the pore constriction

• Disagreement attributed to uncertainty in the diffusioncoefficient and finite ion volume effects – work in progress

MONTE CARLO PARTICLE SIMULATIONMONTE CARLO PARTICLE SIMULATION

• Particle trajectories are resolved in 3D. Electrostatic forces calculatedself-consistently from Poisson equation (P3M)

• Water is assumed to be a continuum background with a givenpermativvity ε. Interaction between ions and water is accounted for bya scattering rate. Flight-times between collisions are generatedstatistically from an average scattering rate. Scattering events areassumed to thermalize the ions.

• Finite size of the ions is accounted for by associating a radius and aLennard-Jones potential to the ions.

initialize ( t = 0 )grid, protein charges

calculate ρfixed

rions→ ρions

solve POISSON

update E

Move ions (E + FLJ)scattering with water

scattering off protein/lipidupdate rions

( ) ( )fixed ionsε φ ρ ρ∇ ∇ = − +i

~

short range forces

~

t → t +dt

Lennard Jones 6-12 potential6 12

4LJ LJ r rσ σϕ ε

= − −

ε

short range repulsion

-0.50

0.00

0.50

1.00

Vol

ts

0.0 2.0 4.0 6.0 8.0 10.0

ion separation (Angstroms)

long rangeweak attraction

P3M

LJ

GRAMICIDIN CHANNELGRAMICIDIN CHANNEL

Na+ trajectories that cross the channel are rare events

CHALLENGESCHALLENGESPoisson→ computational bottleneckSmall ensemble size (1M, 28000 Å3 → ~ 40 ions) Coulomb (N2 = 1600) is muchfaster than Poisson (Nmesh= 28000)How to evaluate Fcoulomb when ions are separated by ε(r)?Use a coarser mesh and include correction for short-range forces?OK for baths – but close to protein need to have dielectric interface resolvedproperly. Coarse mesh also alters the channel cross-section and crossing probability.→ 2 grids (fine grid for defining ion accessible volume, coarser grid for fields)→ non-uniform, non-rectilinear mesh

Channel crossings are rare events (1pA → 6 ions crossing per µs)Extended-ions (spheres) greatly reduces the probability of ion crossing channel –most CPU spend tracking ions that don’t go anywhere near the channel mouth.High ion-water scattering rate 1013 to 1014 Hz, ∆t < 10fs so need about 108 ∆t(1GHz processor, say ∆t → 1s, 1 µs → 1yr)

Boundary conditions – what to do when ion hits electrode? Ion injection fromelectrodes? How far away from the protein do we need to put the electrodes?

Ions are not point charges

Ion-protein/lipid interaction modeled as hard-wall potential – ions withinone ionic radius of protein/lipid surface are scattered randomly.

How is ionic charge distributed on the ion? How is this charge distributionmapped to grid? How to handle the short-range correction?

Test trajectories in prescribed field (PROPHET)bias = 150 mV, 100 mM NaCl, 1µs simulation time

ions modelled as point charges: ~ 25 % current carried by Cl–

ions modelled as “hard” spheres: < 4 % current carried by Cl–

experimentally: Cl– is zero

Simple 1-D PNP (drift-diffusion) solver for ionic-channel

On line at : http: // lipidraft.ncsa.uiuc.edu